4.1.15. Thermodynamics What are examples of path functions and state variables?

  • Path functions: Heat and work (about the boundary)

  • State variables: Internal energy, enthalpy, entropy (about the fluid) What is the First Law of Thermodynamics and what kind of processes is it applicable to?

\[\text{Net heat added to system} + \text{Net work done on system} = \text{Increase in internal energy of system}\]
\[dq + dw = de\]

(lower case implies per unit mass)

The first law applies to reversible and irreversible processes What is the Second Law of Thermodynamics and what is it for a reversible process?

\[\text{Gross heat added to system} + \text{Net work done by system} \ge 0\]
\[dq + \Sigma = Tds\]

i.e. there is irreversible work done by internal molecular changes \(\Sigma\)

For a reversible process \(\Sigma = 0\)

i.e. \(ds = {dq \over T}\) What is Enthalpy?

From the \(1^{st}\) law, for both reversible and irreversible processes:

\[dq = d(e+pv) = dh\]

where \(dw = -d(pv)\)

Enthalpy is a state variables

For both reversible and irreversible processes it equals the internal energy plus pressure-volume potential energy. What is Entropy?

From the \(2^{nd}\) law for a reversible process:

\[s_2 - s_1 = \int_1^2 {dq \over T}\]

Entropy is a macro state variable

For a reversible process it is the infinitisimal heat added over constant temperature (non-adiabatic). What are the two property relations for entropy change from the 1st and 2nd laws of thermodynamics at constant pressure and constant volume?

  • 1st Law (reversible and irreversible) \(\longrightarrow de = dw+dq\)

  • Reversible work done by system \(\longrightarrow dw = -pdv\)

  • Reversible heat added to system (2nd law) \(\longrightarrow dq = Tds\)

  • For reversible and irreversible processes (1) \(\longrightarrow Tds = de + pdv\)

  • Definition of enthalpy \(\longrightarrow dh = d(e+pv) = de + d(pv) = de + pdv + vdp\)

  • By rearrangement \(\longrightarrow de = dh - pdv - vdp\)

  • For reversible and irreversible processes (2) \(\longrightarrow Tds = dh - vdp\) What is the entropy change for an ideal gas for a reversible and irreversible process?

  • \(ds = {de \over T}+{p \over T}dv\)

  • \(ds = {dh \over T} - {v \over T}dp\)

  • Calorically perfect

\(\longrightarrow de=c_vdT\)

\(\longrightarrow dh=c_pdT\)

  • Ideal gas

\(\longrightarrow p = {RT \over v}\)

\(\longrightarrow v = {RT \over p}\)

  • Hence

\(\longrightarrow s_2 - s_1 = c_v \int_1^2 {dT \over T} + R \int_1^2 {dv \over v} = c_v ln{T_2 \over T_1} + R ln{v_2 \over v_1} = c_v ln{T_2 \over T_1} + R ln{\rho_1 \over \rho_2}\)

\(\longrightarrow s_2 - s_1 = c_p \int_1^2 {dT \over T} - R \int_1^2 {dp \over p} = c_p ln{T_2 \over T_1} - R ln{p_2 \over p_1}\) How are the isentropic relations for an ideal gas derived from the entropy change?

  • Isentropic \(\longrightarrow s_2 - s_1 = 0\)

  • Hence

\(\longrightarrow 0 = c_v ln{T_2 \over T_1} - R ln{\rho_2 \over \rho_1}\)

\(\longrightarrow 0 = c_p ln{T_2 \over T_1} - R ln{p_2 \over p_1}\)

  • Definition of \(c_v\) and \(c_p\):

\(c_v = {R \over {\gamma -1}}\)

\(c_v = {{\gamma R} \over {\gamma -1}}\)

  • Hence:

\(\longrightarrow ln{\rho_2 \over \rho_1} = {1 \over {\gamma -1}} ln{T_2 \over T_1}\)

\(\longrightarrow ln{p_2 \over p_1} = {{\gamma} \over {\gamma -1}} ln{T_2 \over T_1}\)

  • Hence:

\(\longrightarrow {\rho_2 \over \rho_1} = {T_2 \over T_1}^{1 \over {\gamma -1}}\)

\(\longrightarrow {p_2 \over p_1} = {T_2 \over T_1}^{{\gamma} \over {\gamma -1}}\)

  • Hence:

\(\longrightarrow {p_2 \over p_1} = {\rho_1 \over \rho_2}^{\gamma}\)

\(\longrightarrow {p \over {\rho^{\gamma}}} = \alpha\)